If tan θ = 4/9 and cot ϕ = 7/5 , find the exact value of sin ( θ + ϕ ) .

Question

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If tan θ = 4/9 and cot ϕ = 7/5 , find the exact value of sin ( θ + ϕ )

You do not need to simplify any radicals.

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Answer 1

The sin(a+b) = sin(a)cos(b)+cos(a)sin(b). So all that needs to be determined is the value of each. Whenever a trig value is known, all six are known (sin, cos, tan, cot, sec, csc). Draw a right triangle for the first one. Label the corner angle as theta. Since tan is opposite / adjacent. Tan(θ) = 4/9, by Pythagorean theorem, the hypotenuse must be sqrt(16+81) or sqrt(97). So sin(θ) = 4/sqrt(97). Similarly, cos(θ) = 9/sqrt(97). Now do the same for the other triangle. If cotΦ = 7/5, hyp = sqrt(74), so sin(Φ) = 5/sqrt(74) and cos(Φ) = 7/sqrt(74). Now do the math according to formula. Key point here is that whenever you know one trig value, you know them all.